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Transcript
```November 30, 2010
2.6 Related Rates
Guidelines for Solving Related-Rates Problems
1. Identify all given quantities and quantities to
be determined. Make a sketch and label the
quantities.
2. Write an equations involving the variables
whose rates of change either are given or are
to be determined.
3. Using the Chain Rule, implicitly differentiate
both sides of the equation with respect to
time t.
4. Substitute into the resulting equation all
known values for the variables and their rates
of change. Then solve for the required rate
of change.
1
November 30, 2010
Section 3.1 Extrema on an Interval
Definition of Extrema
Let f be defined on an interval I containing c.
1. f(c) is the minimum of f on I if f(c) ≤ f(x)
for all x in I.
2. f(c) is the maximum of f on I if f(c) ≥ f(x)
for all x in I.
A min or max of f on I is called the extremum.
Extreme Value Theorem
If f is continuous on [a, b], then f has both a
min and max on [a, b].
Definition of Relative Extrema
1. If there is an open interval containing c on which
f(c) is a maximum, then f(c) is called a relative
maximum of f, or we say "f has a relative max at
2. If there is an open interval containing c on which
f(c) is a minimum, then f(c) is called a relative
minimum of f, or we say "f has a relative min at
2
November 30, 2010
Definition of a Critical Number
Let f be defined at c. If f '(c) = 0 or if f is not
differentiable at c, then c is a critical number
of f.
Theorem 3.2
If f has a relative min or max at x = c, then c
is a critical number of f.
Guidelines to find the extrema of a continuous
function f on [a, b]
1. Find the critical numbers of f in (a, b),
2. Evaluate f at each critical number in (a, b),
3. Evaluate f at each endpoint of [a, b],
4. The least of these values in the min. The
greatest is the max.
3
November 30, 2010
Rolle's Theorem
Let f be continuous on [a, b] and differentiable
on (a, b). If f(a) = f(b), then there is at least one
number c in (a, b) such that f '(c) = 0.
f(b) = f(a)
a
b
The Mean Value Theorem
If f is continuous on [a, b] and differentiable
on (a, b), then there exists a number c in (a, b)
such that
f(b) − f(a)
f '(c) =
b−a
f(b)
f(a)
a
b
4
```